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What I am asking for is if there is any theory related to this real series $f(s)=\sum_{n=1}^\infty e^{-n^s}$ and $s\ge 1$.

As far as I know, if $s = 1$, it's a simple geometric series, and if $s = 2$, the series can be converted to one case of Riemann Theta Function. So what if $s$ is generalized to any real number larger than or equal to 1?

Eddie Lin
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    It's analytic continuation is a bit complicated and was discussed in https://math.stackexchange.com/questions/3637584/analytic-continuation-of-phis-sum-n-ge-1-e-ns/3808724#3808724 – reuns Apr 03 '22 at 09:53
  • @reuns Thank you so much! That's super helpful. – Eddie Lin Apr 09 '22 at 21:50

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